Method and apparatus for controlling a drive in a machine tool

ABSTRACT

A method and apparatus for controlling the output to a drive for moving a workpiece and/or tool along a defined path in a machine tool. The drive output can be modified according to a periodic controlling variable or a periodic disturbance variable. The defined path of the workpiece is divided into equidistant subdivisions and a control deviation corresponding to each subdivision of the defined path is determined and stored. A control variable is derived from the control deviation and utilized to control the drive output.

FIELD OF THE INVENTION

The invention relates to a method for controlling the controller outputfor a drive for moving a workpiece and/or tool along a defined path in amachine tool, wherein the controller output can be modified according toa periodic controlling variable or a periodic disturbance variable inresponse to a force acting upon the movement, and to a controlarrangement for the application of the method.

BACKGROUND OF THE INVENTION

A control arrangement is known from the German utility patent G 92 00708.2, which allows for periodically changing controlling anddisturbance variables. Processes comprising these variables includestock removal operations such as planing, turning, milling, grinding,boring etc. The periodicity relates not only to the controllingvariables, such as the change of the angular position of lathe work, butalso to disturbance variables, for example the cutting forces.Additionally, the utility model relates to the coupling of a periodiclinear motion with a periodic rotary motion, the coupling of two rotarymotions or the coupling of periodic linear motions. Moreover, the priorcontrol setup relates also to fluctuations of synchronism in machinetool drives due to a pole reversal. A tool or workpiece is considered tobe a work object.

Conventional control methods do not allow a sufficient suppressing ofexternal disturbance variables thus inviting the risk of instabilities.This relates notably also to the case of equal periodicity of thecontrolling variable and the disturbance variable.

Described in the book O. Follinger "Regelungstechnik," Huthig-Verlag,Heidelberg, 6. edition, p. 519 with literature references p. 526, is anextension, proposed by C. Johnson, of a so-called Luenberger observer bya disturbance model. In this context, a disturbance estimate, calculatedcorrectly by phase and amplitude, is with inverted sign superimposed onthe controlled system, such as is done similarly in the known case ofdirectly measurable disturbances by feeding the disturbance variablesforward into the controller. This method is suited favorably in using adigital process control computer. For each scanning step, however, boththe mathematical mapping of the controlled system and of the disturbancemust be calculated. As the complexity of the disturbance modelincreases, the continual calculation leads to appreciable calculatingeffort, which in the case of fast processes, such as in controllingelectrical drives in machine tools, can overwhelm typical standardmicroprocessors. The scanning intervals can become impermissibly high,which invites the risk of instabilities. With the addition of a digitalmeasuring system, such as positional determination by angle transmitter,control can be rendered impossible when fluctuations assuming higherfrequencies due to the low scanning frequency are being folded back,since the scanning theorem is violated. In addition to eliminatingdisturbances, the adjustment of a periodic controlling variable which isfree of contouring error also poses difficulties.

The aforementioned German utility model G 92 00 708.2 provides asolution to the described control task which does not require adisturbance variable observer. The solution consists in the fact thatfor adjustment of the periodic controlling variable and/or for tune-outof a periodic disturbance variable, the controller features at least oneconjugate complex pole pair, wherein the amount of each of the two polesis equal or approximately equal to the periodic frequency of thedisturbance variable. In practice, the configuration of the feedbackloop is such that a disturbance compensation system in the form of adisturbance controller is wired parallel to a controller configured inaccordance with conventional aspects. This disturbance controller hasthe aforementioned properties, which are described by the behavior of anintegral band-pass controller or a proportional band-pass controller.

The properties of both controller types are represented by their complextransfer functions, that is, the output of the disturbance controller,based on an input that matches the control difference, is represented bythe Laplace transform of the respective variables. The complex functionvariable s allows illustration by way of the relation s=δ+τω in afrequency representation that is determined by the amplitude responseand the phase response of the controller. Besides, an inverted Laplacetransformation makes it possible to achieve a direct representation inthe time range.

If in the prior control setup the frequency of the periodic motionchanges, the frequency of the disturbance controller of the disturbancecompensation system must be adapted. Problems arise when such frequencychanges occur within the original period.

Therefore, the objective underlying the invention is to propose animproved control method that eliminates the disadvantages describedabove and, notably, allows reacting directly to frequency changes of aperiodic controlling variable or of a periodic disturbance variablewhich act upon the movement of a work object being moved along aspecific local coordinate.

Furthermore, a control setup for realization of the inventional controlmethod is proposed.

SUMMARY OF THE INVENTION

The present invention comprises a control method in which thepredetermined or defined path that a workpiece travels is subdivided inequidistant sections, and where for the equidistantly subdivided definedpath of the work object the appropriate control deviation is determinedand stored. A corresponding control variable is subsequently obtainedfrom the control deviation, depending on the momentary position of thework along the defined path.

The invention is based on the premise that periodic reference variablesand disturbance variables occur frequently not in the originaldependence on time, but as a function of a defined path, for instance aposition or an angle. By inferring the control deviation on the groundsof the determined position of a work object and obtaining from thecontrol deviation a control variable for the drive of the work it ispossible to achieve considerably improved control properties.

In a preferred embodiment the inventional control method is governed bya process control computer.

The value of the current controller output is determined from the valueof the current control deviation and of stored, recent values of thecontrol variable and the control deviation. Only the two precedingvalues of the control deviation and of the control variable arepreferably needed for that purpose, so that the past values of thecontrol variable and of the control deviation need to be stored for onlya part of a period of the periodic reference variable or the periodicdisturbance variable, which act upon the movement of the work.

The present invention also comprises a control setup for controlling thedrive for a work object to be moved along a specific defined path, whichsetup comprises a device for subdividing the path of the workpiece intoequidistant sections, for example an incremental angle transmitter, withthe controller of the control setup or part thereof generating thecontrol variable from the control deviation presented to it whichdepends on the momentary position of the work along the defined path oftravel. The controller comprises, besides the part controlling in thelocal domain, a controller part that is arranged parallel and controlscontingent on time.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a block diagram of a prior-art control arrangement.

FIG. 2 shows a block diagram of a control setup according to the presentinvention.

FIG. 3 schematically illustrates a shaving grinder for explanation ofthe present invention.

FIG. 4 shows a diagram illustrating the angle error occurring with agear grinder within a period of the work object.

FIG. 5 shows an illustration of a sine-shaped error of the angle ofrotation, with a fluctuating period in the time range.

FIG. 6 shows an illustration of the same sine-shaped error of the angleof rotation as in FIG. 5 in the local range with a constant period.

FIG. 7 illustrates a gear cutting machine operating in a continuousrolling method with the control method of the present invention applied.

FIG. 8 illustrates a circuit arrangement where the control deviation isobtained from a variable other than the defined path of the work.

FIG. 9 shows a schematic illustration of the oscillation model formathematical description of a disturbance variable controller.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

The invention is described hereafter with reference to the drawings withthe aid of exemplary embodiments.

The control arrangement shown in FIG. 1 is structurally a known feedbackloop with a controller consisting of two components wired parallel andwith a controlled system 2. Derived from the controlling variable w andthe output y of the controlled system, by differentiation, is thecontrol deviation e, which is applied across the input of the twocontroller parts 1 and 3 wired parallel. The outputs u_(R) and u_(M) ofthe two controller parts are added, and the result is the correctionvariable for the controlled system. The disturbances acting on thecontrolled system are signified by z in FIG. 1 and they act on thecontrolled system 2 as additional input and affect the actual correctionvariable u. As described already in the Background of the Invention,such a control arrangement is previously known from the German utilitymodel G 92 00 708.2.

Periodic reference and disturbance variables occur which are notdependent on time. Rather, an original dependence is given as a functionof the location on a specific predetermined or defined path, forexample, a dependence upon a position or an angle. This dependenceoccurs, for example, due to pole reversal of an electric drive, whichleads to an irregularity of the movement. In this case, disturbances arecreated depending on event. Disturbances contingent on a specific localposition occur as well in a gear grinder where an indexing error occursafter a complete revolution, and in the case of a gear shaver where thegrinding wheel touches the work always in the same position of theslide.

The invention allows for this circumstance by determining the controldeviation caused by locally dependent disturbances contingent on theposition of a workpiece as it travels along a defined path, that is, thecontrol difference is not related to time, but to the position of thework. To that end, the position of the work along the specific definedpath is inventionally monitored during the control procedure. Thus,instead of a strictly time-dependent control deviation, aposition-dependent control deviation is used in performing the control,that is, a control deviation in the local range. A control variable forthe drive of the work is subsequently obtained from the controldeviation.

The consequence for the prior-art control arrangement shown in FIG. 1is, as illustrated in FIG. 2, that it must be modified. According to thepresent invention, the controller comprises additionally two controllerparts 1, 3 whereof the first controller part 1 now controls in the timedomain and the second controller part 3 in the local domain. That is,the feedback to the first controller part is the time-dependent controldifference e(t) and to the second controller it is theposition-dependent control deviation, presently the angle-dependentcontrol deviation e(φ) as input variable. Hence, a transition from thecontrol in the time domain to control in the local domain is effected inthe second controller part 3. The first controller part 1, with thetime-dependent control difference e(t) as input variable, now servesmerely stationary control procedures for the case that the position ofthe work on the defined path has not changed yet or will change nolonger. This may be the case, for example, in the cut-in, or startup, ofthe drive of the work to be machined. If only the controller part 3controlling in the local domain were present, no control procedureswhatsoever would take place during an operational phase with anexclusive control in the local domain, since there would be no travelalong the defined path during this phase of operation. To avoid this,the controller part 1 (controlling exclusively in the time domain andfeaturing as input variable the time-dependent control deviation e(t))is wired parallel to the controller part 3.

The invention and its effects are illustrated hereafter with the aid ofthe gear shaver shown in FIG. 3.

A slide 7 is arranged on sliding base 6 in a fashion allowing linearmovement. The linear displacement of slide 7 is effected by drive 9 vialead screw 8, in a way such that the slide reciprocates as indicated bydouble arrow 10. Mounted on slide 7, rotatable about axis 22, is theshaving cutter 11 to be ground. For simplicity, only two teeth 12 areshown of shaving cutter 11. The shaving cutter can be rotationallyreciprocated by means of drive 14 via worm gear mechanism 15, asindicated by double arrow 13. Grinding wheel 19 engages the spacebetween two teeth 12 of shaving cutter 11, the grinding wheel rotatingabout skewed axis 20 as indicated by arrow 21. Each reciprocating motionof shaving cutter 11 coincides with an abrasive contact with thegrinding wheel 19. This contact has a disturbing effect on the torque ofdrive 14, by which shaving cutter 11 must be rotated back and forth as aconsequence of the linear motion of slide 7. This disturbance may resultin defects in grinding the shaving cutter 11. As described above, suchdisturbance occurs at each reciprocating motion of shaving cutter 11 asit enters in abrasive contact with grinding wheel 19. Since thereciprocating motion of slide 7 and the reciprocating motion of shavingcutter 11 occur periodically, the disturbance contact between shavingcutter 11 and grinding wheel 19 also occurs at the same basic frequencyas the movement of slide 7. Hence, periodic and locally dependentdisturbances are concerned in this case, since the disturbances occuronly in a specific slide position.

According to the present invention, the dependence of the torque errorof drive 14 upon the slide position l is now determined. Depending onthe change Δl of the slide position, the respective control deviation eis determined, and controller 4 generates a control variable u.Required, therefore, is feeding to controller 4 the slide positionprovided by sensor 16. In the exemplary embodiment illustrated in FIG.3, controller 4 comprises not only the part 3 that controls in positiondependence but, wired in parallel, also a part 1 controlling intime-dependent manner. As described with the aid of FIG. 2, theposition-dependent control deviation is fed to controller part 3, thetime-dependent control deviation to part 1. Thus, controller 4 producesfrom the slide position and the control deviation depending on it thecontrol variable u for drive 14 of shaving cutter 11. Since the abrasivecontact between grinding wheel 19 and shaving cutter 11 diminishes asthe grinding operation continues, the disturbing effect of the abrasivecontact on the torque of drive 14 diminishes with the ongoing grindingoperation. Therefore, continual feedback of the control deviation e tocontroller 4 is necessary for detection of the momentary dependence ofthe control deviation e on the position of slide 7.

As another example for illustration of the invention, a gear grinder isconsidered where within one revolution of the gear the indexing error foccurs, which is illustrated in FIG. 4 and depends on the angularposition φ of the gear. Indexing error f passes within one revolution ofthe work exactly through one period. Based on the dependence of indexingerror f upon angular position φ, an appropriate control variable isgenerated for tune-out of the indexing error. Although the basic patternof dependence of angular error f upon angular position φ of the gearremains unchanged, the dependence will nonetheless change quantitativelywith increasing machining, to the effect that the indexing errordiminishes gradually contingent upon the angular position of the gear.It is conceivable to determine, after each constant angle of rotation Δφof the gear, its angular position φ along with the value of indexingerror f depending on the momentary angular position. This methodsuggests itself notably for computerized control methods employingdiscrete scanning values. Inventionally, the control is carried out notin the time domain, but in the local domain, so that the scanning valuesaccording to the invention are not obtained in equidistant timeintervals, but in equidistant angular intervals Δφ. The angle ofrotation 2π as illustrated in FIG. 4, f=f(φ), corresponds in timerepresentation to f=f(t) of one period T.

The effect of the inventional measure is illustrated hereafter withreference to FIGS. 5 and 6. FIG. 5 shows the pattern of a periodic andtime-dependent rotational angle error e(t) with a variable period in thetime domain. From FIG. 5 it follows that due to the changing period ofthe time-dependent rotational angle error e(t) changes occur also in theindividual period of the oscillations, with the result that fluctuatingperiods T_(t1) through T_(t5) occur. The same periodically occurring,sine-shaped rotational angle error is illustrated in FIG. 6 in the localdomain, that is, with reference to FIG. 4 depending on the angularposition φ of the gear. Due to the fact that the angular error e(φ) 2πoccurs periodically, a period deviation in the local domain, that is, asregards the angular position, does not occur even with a frequencychange of the rotational angle error in the time domain, for instance,by acceleration of the gear drive. Therefore, the periods T.sub.φ1through T.sub.φ5 of the locally dependent rotational angle error e(φ)appear constant in the local domain.

FIG. 7 shows a further example for illustration of the presentinvention. A gear type, or worm type, tool 45 rolls continuously with agear-shaped workpiece 42. Workpiece 42 is speed-controlled by drive 43,producing the direction of rotation indicated by arrow 44. Tool 45 isdriven by sequential drive 46, producing the direction of rotationindicated by arrow 47. Coupled with work 42 is an angle transmitter 49and coupled with tool 45 is a pulse transmitter 48 and angle transmitter50. The output signals generated by the two angle transmitters 49, 50are transmitted to the difference former 51. The control deviationgenerated thereby is passed to angle controller 52 producing the setspeed for sequential drive 46. The set speed w and the speed controlleroutput x generated by pulse transmitter 48 are passed to anotherdifference former 53, which generates the control deviation e. Tool 45has in comparison to work 42 small teeth deviations, thereby causingspeed and angle fluctuations between tool 45 and work 42. Tool drive 46is governed by controller 56 with feedback of correction variable u totool drive 46. According to the invention, controller 56 comprises inaddition to the known time-dependent controller part 1 a controller part3 with a position-dependent mode, which receives as input variable theposition-dependent control deviation e(φ). Both controller parts 1, 3are wired parallel. The angular position of the work is detected byangle transmitter 49, preferably after equidistant variations Δφ, andthe control deviation corresponding to the current angular position isdetermined by difference former 53.

In the embodiments shown in FIGS. 3 and 7, the control deviation epassed to controllers 4 and 56, respectively, derives directly from therespective defined path, that is, respectively, from the position ofslide 7 and the angular position of work 42. To show that the controldeviation e can be derived also from a variable other than the definedpath of the respective work, FIG. 8 shows a further, simplifiedexemplary embodiment. An elliptic disk 60 is machined with the aid ofcutter 67. Work drive 61 rotates the work 60 about axis 63 in thedirection of arrow 62. The task is rounding the work 60 with the aid ofcutter 67. The thickness of the stock removed by the cutter increaseswhenever, as shown in FIG. 8, the part of disk 60 with the semi-majoraxis passes by cutter 67. Each contact between disk 60 and cutter 67affects the drive of disk 60, producing a rotational angle errordepending on the respective angular position of disk 60. The angularposition of disk 60 is determined by angle transmitter 64 with feedbackto controller 66. Mounted in flexible manner, cutter 67 swingsperiodically about a set position due to the elliptic shape of disk 60.The feed of cutter 67 dictates to what extent disk 60 is still ellipticin shape or whether it has assumed circular shape. Therefore, theposition l of cutter 67 is determined by position transmitter 65, andthe control deviation e is obtained by the difference between areference variable w and the feed path l of cutter 67, which differenceis passed to controller 66. In the exemplary embodiment illustrated inFIG. 8, control is thus performed in the local domain, since a controlcorrection variable u for the cutter drive 68 is determined depending onthe angular position φ of disk 60 and the control deviation e which isobtained not from the angular position p of disk 60, but from anothervariable, namely position l of cutter 67. φ The control method accordingto the invention is preferably applied with the aid of a digitalcomputer, using an algorithm that will be developed hereafter.

The calculation of the coefficient of a control algorithm operating inthe local domain will be shown hereafter.

The known oscillation model may, with reference to FIG. 9, be describedgenerally by a sine-stimulated mass m with a spring constant c anddamping constant d. The input variable of the oscillation model is thecontrol deviation e, which may be perceived as the force stimulating themass m. The output of the oscillation model corresponds to the velocityof the oscillating mass. Therefore, the following known differentialequation can be established for the velocity u_(M) :

    m·u.sub.M +d·u.sub.M +c·u.sub.M =e(1)

Transferring the differential equation in the time domain by means ofLaplace transformation to the frequency domain, the following formularesults as the transfer function of the disturbance controller: ##EQU1##

Here, h₁ is analogous to the damping of the mass oscillator, h₂ allowsfor the squared reciprocal of the oscillation number from 0 to 2π, andg₁ determines the amplitude amplification of the input signal. When now,depending on the position of the work, disturbances occur on a definedpath, the correction variable and the control deviation of the controlsetup (as described with the aid of FIG. 6 already) will no longer havean original dependence on frequency. This means that the complexvariable s in the pertaining transfer function of the oscillation modelhas no longer the dimension rad⁻¹, but is dimensionless.

Equation (2) describes the relationship between control deviation andoutput of the disturbance controller as a system of second order.However, disturbance controllers of systems of higher order can berealized as well.

According to the book "Grundlagen der Regelungstechnik" Fundamentals ofControl Engineering by Dorrscheidt and Latzel, Teubner-Verlag,Stuttgart, 1989, p. 428-429, control algorithms in polynomial form couldbe proposed for disturbance controllers of second order, whichalgorithms are used in converting control methods to digital computer.Generally applicable for the transfer function of a second-ordercontroller with input e_(k) and output u_(mk) is thus: ##EQU2##

For the transfer function described under (3), the following controlalgorithm is available according to Dorrscheidt/Latzel:

    u.sub.m.sbsb.k =d.sub.0 ·e.sub.k +d.sub.1 ·e.sub.k-1 +d.sub.2 ·e.sub.k-2 +c.sub.1 ·u.sub.M.sbsb.k-1 +c.sub.2 ·u.sub.M.sbsb.k-2                                (4)

According to Dorrscheidt/Latzel, the coefficients of the controlalgorithm (d₀, d₁, d₂, c₁, c₂) can be determined depending on thecoefficients of the pertaining transfer function (g₀, g₁, g₂, h₀, h₁,h₂). The assumption with algorithm (4) is that the discrete values ofthe control deviation e_(k-i) and of the control correction variableum_(k-i) are obtained by scanning at an interval T. Indices k, k-1, k-2represent the current value and the input or output value of thedisturbance controller preceding the current value by T or 2T.

The formulas derived from the said book for calculating the coefficientof the aforementioned algorithm for control in the time domain can betransferred easily to the local domain, by replacing the scanninginterval T by the scanning interval in the local domain. As describedwith the aid of FIG. 4, the position of the work on the defined path,the value of the control deviation and the dependence of the controldeviation upon the position of the work on the defined path aredetermined after each equidistant change Δl of the work position on thedefined path. The coefficients of the control algorithm according toformula (4) are thus subject to the following: ##EQU3##

In case the defined path is the angle of rotation φ, a scanning intervalof Δφ applies. Basing on the transfer function according to formula (2)(i.e., g₀ =g₁ =0 and h₀ =1 in the transfer function according to formula(4)), as derived from the oscillation model, the following applies tothe locally (angularly) dependent coefficients of the control algorithm:##EQU4##

u_(mk) describes now the control correction variable issued by thedisturbance variable controller for a current angle, that is, the outputof the disturbance variable controller, while e_(k) describes thecurrent input signal, or current control deviation, of the disturbancevariable controller which, modeled on FIG. 4, is caused by therotational angle error f. Indices k, k-1, k-2 are now representative of,respectively, the current angular position φ and the input or outputvalues of the disturbance variable controller which precede the currentangular position φ by Δφ or 2Δφ. Noteworthy for the control algorithmaccording to formula (4) is that the value of the current controlcorrection variable u_(mk) can be calculated from merely the twopreceding values of the control correction variable u_(m) and thecontrol deviation e as well as the momentary value of the controldeviation. For the inventional control method and the inventionalcontrol setup this means that with a second-order system only the lasttwo values of the control deviation e_(k-1), e_(k-2) preceding thecurrent angular position and of the control correction variableu_(mk-1), u_(mk-2) need to be stored. This has the appreciable advantagethat the values of control deviation e required for calculating thecurrent value of control correction variable u_(mk) and the precedingvalues of the control correction variable u_(M) need to be stored notover at least one period, but (as evident with the aid of FIG. 4) onlyfor a fraction of a period. Hence, the disturbance variable controlleris able to respond very quickly to a change of phase and amplitude of asine-shaped disturbance variable.

Since the aforementioned control algorithm does not require thecalculation of the system, or system model representing a mathematicaldescription of the controlled system, the computing effort remainswithin limits. Therefore, the problems mentioned with respect to theLuenberger observer cannot arise. The inventional control method andarrangement thus allows a control that is dependent on part of a period,since the values necessary for calculation are being stored only for afraction of the period, namely only the values preceded by Δφ and 2Δφ.

As mentioned already, the remaining part of the controlled system mustnonetheless be described and controlled depending on time, despite thecontrol in the local range which is dependent on the position of thework on a defined path. As illustrated already with the aid of thepreceding exemplary embodiments, it is therefore necessary to determineafter each completion of an angle Δφ the rotational angle deviationf(φ), respectively the control deviation e(φ). This can be accomplishedwith the aid of a timer interrupt and with a further interrupt to be setup in the process control computer used in the control setup.

We claim:
 1. A method for controlling the drive of a first work objectof a machine tool, where the first work object interacts with a secondwork object which travels along a controlled path with a motion that isaffected by said interaction with said first work object, comprising thesteps of:generating a system deviation signal as a function of thedifference between a reference signal and a signal representing themotion of said first work object; supplying said system deviation signalto a controller that produces a drive signal for controlling said drivesuch that said first work object is moved in a manner that minimizessaid system deviation signal; sensing the motion of said second workobject and producing sampling pulses at predetermined points along saidcontrolled path of said second work object; utilizing said samplingpulses in said controller to sample the value of said system deviationsignal at each of said predetermined points; successively calculatingposition-related values of said drive signal using in each calculationthe most recently sampled value of said system deviation signal; andapplying said position-related drive signals to said drive to controlthe movement of said first work object.
 2. The method of claim 1comprising the further steps of:successively calculating time-relatedvalues of said drive signal using in each calculation the present valueof said system deviation signal supplied to said controller; andapplying said time-related drive signals to said drive in the absence ofsaid sampling pulses to control the movement of said first work object.3. The method of claim 1 wherein said predetermined points along saidcontrolled path of said second work object are equidistant from oneanother.
 4. The method of claim 1 wherein said successiveposition-related drive signals are calculated according to a controlalgorithm of a specific order under control of a process computer. 5.The method of claim 1, comprising the further steps of:storing saidsampled values of said system deviation signal; calculating a new valueof said position-related drive signal each time a new sampled value ofsaid system deviation signal is obtained; and storing successivecalculated position-related drive signals, said control algorithmutilizing each new sampled value of said system deviation signal as wellas said stored system deviation signals and stored position-relateddrive signals to determine said new value of said position-related drivesignal.
 6. The method of claim 5 wherein said control algorithm is ofthe second order based on the transfer function ##EQU5## according tothe following formula: ##EQU6## where: Δl=equidistant change inworkpiece position;l_(k) =workpiece position; l_(k-1) =precedingworkpiece position; e_(k) =control deviation; e_(k-1) =first precedingcontrol deviation; e_(k-2) =second preceding control deviation; u_(mk)=control variable; u_(mk-1) =first preceding control variable; u_(mk-2)=second preceding control variable; g₀, g₁, g₂, h₀, h₁, h₂ =coefficientsof the transfer function; s=complex variable of transfer function.
 7. Amachine tool in which first and second work objects are moved in acontrolled manner and interact with one another to modify one of saidobjects, and in which said second work object travels along a controlledpath with a motion that is affected by said interaction with said firstwork object, said machine tool comprising:a drive for moving said firstwork object; means for generating a reference signal; means forgenerating a motion signal representing the motion of said first workobject; a feedback circuit for generating a system deviation signal as afunction of the difference between said reference signal and said motionsignal; a controller receiving said system deviation signal andproducing in response thereto a drive signal for controlling said drivein a manner that minimizes said system deviation signal; a sensor forsensing the motion of said second work object and providing samplingpulses at predetermined points along said controlled path of said secondwork object, said controller operating in response to each of saidsampling pulses to sample the value of said system deviation signal ateach of said predetermined points; and a calculator included in saidcontroller for successively calculating position-related values of saiddrive signal using in each calculation the most recently sampled valueof said system deviation signal, said position-related drive signalsbeing applied to said drive to control the movement of said first workobject.
 8. The machine tool of claim 7, further comprising:meansincluded in said calculator for successively calculating time-relatedvalues of said drive signal using in each calculation the present valueof said system deviation signal received by said controller, saidtime-related drive signals being applied to said drive to control themovement of said first work object in the absence of said samplingpulses.
 9. The machine tool of claim 7 wherein said predetermined pointsalong said controlled path of said second work object are equidistantfrom one another.
 10. The machine tool of claim 7 wherein saidsuccessive position-related drive signals are calculated according to acontrol algorithm of a specific order under control of a processcomputer.
 11. The machine tool of claim 6 wherein said calculatorcalculates a new value of said position-related drive signal each time anew sampled value of said system deviation signal is obtained, andwherein said controller includes:first storage means for storing saidsampled values of said system deviation signal; and second storage meansfor storing said successive calculated values of said position-relateddrive signals, said calculator utilizing each new sampled value of saidsystem deviation signal as well as said stored system deviation signalsand stored position-related drive signals to determine the new value ofsaid position-related drive signal.
 12. The machine tool of claim 11,wherein said control algorithm is of the second order based on thetransfer function ##EQU7## according to the following formula: ##EQU8##where: Δl=equidistant change in workpiece position;l_(k) =workpieceposition; l_(k-1) =preceding workpiece position; e_(k) =controldeviation; e_(k-1) =first preceding control deviation; e_(k-2) =secondpreceding control deviation; u_(mk) =control variable; u_(mk-1) =firstpreceding control variable; u_(mk-2) =second preceding control variable;g₀, g₁, g₂, h₀, h₁, h₂ =coefficients of the transfer function; s=complexvariable of transfer function.